Answered

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In 1992, South Dakota's population was 10 million. Since then, the population has grown by 1.4% each year. Based on this, when will the population reach 20 million?

Sagot :

fieryanswererft

Answer:

  • 49.9 years or 50 years later.
  • At year : 2042

Explanation:

use compound interest formula:  [tex]\sf \boxed{ \sf P ( \sf 1 + \dfrac{r}{100} )^n}[/tex]

[tex]\rightarrow \s \sf 10( 1 + \dfrac{1.4}{100} )^n = 20[/tex]

[tex]\rightarrow \sf ( 1.014) ^n = 2[/tex]

[tex]\rightarrow \sf n( ln( 1.014) ) = ln(2)[/tex]

[tex]\rightarrow\sf n = \dfrac{ln(2)}{ln( 1.014)}[/tex]

[tex]\rightarrow\sf n = 49.8563 \ years[/tex]

semsee45

Answer:

General form of an exponential equation:  [tex]y=ab^x[/tex]

where:

  • a is initial value
  • b is the base (or growth factor in decimal form)
  • x is the independent variable
  • y is the dependent variable
  • If b > 1 then it is an increasing function
  • If 0 < b < 1 then it is a decreasing function
  • Also b ≠ 0

Given information:

  • initial population = 10 million
  • growth rate = 1.4% each year

⇒ growth factor = 100% + 1.4% = 101.4% = 1.014

Inputting these values into the equation:

[tex]\implies y=10(1.014)^x[/tex]

where y is the population (in millions) and x is the number of years since 1992

Now all we need to do is set y = 20 and solve for x:

[tex]\implies 10(1.014)^x=20[/tex]

[tex]\implies 1.014^x=2[/tex]

[tex]\implies \ln 1.014^x=\ln 2[/tex]

[tex]\implies x\ln 1.014=\ln 2[/tex]

[tex]\implies x=\dfrac{\ln 2}{\ln 1.014}[/tex]

[tex]\implies x=49.85628343...[/tex]

1992 + x = 2041.8562....

Therefore, the population will reach 20 million during 2041, so the population will reach 20 million by 2042.

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